I have some understanding that with qubits, I can represent a $2^n$ size vector using only $n$ qubits. However, I'm having trouble putting this together in a way that can make a useful circuit. Say I want to represent the input $(0 \ 0 \ 0 \ 1)$ for my circuit.
I should be able to do this with only 2 qubits. Starting with the initial state $|0\rangle|0\rangle$, which as far as I know represents the vector $(1 \ 0 \ 1 \ 0)$, I can use a not gate on the second qubit to transform that to $(1 \ 0 \ 0 \ 1)$. However, I'm not sure what operation I can do to put the first qubit into a $(0 \ 0)$ state.
On top of that, I have even less of an idea of how to represent real-value vectors, like $(0.5 \ 0.5)$ (assuming including a normalization constant). I have a feeling I can use phase, but I'm not sure how to set up a circuit to achieve this.
Edit: Realized my earlier representation was very off. The initial state $|00\rangle$ is the same as the vector $(1 \ 0 \ 0 \ 0)$, so transforming that into the the desired $(0 \ 0 \ 0 \ 1)$ Would just require two NOT gates:
0 -- NOT -- 1 0 -- NOT -- 1
However, I'm still unsure how to represent real-valued vectors, or even a vector like $(1 \ 0 \ 1 \ 0)$.